Let $\mathscr{ F}$ be a formation and $G$ a finite group. The weak norm of a subgroup $H$ in $G$ with respect to $\mathscr{ F}$ is defined by $N_{ \mathscr{ F}}(G,H)=\underset{ T\leq H}{ \bigcap}N_G(T^{ \mathscr{ F}})$. In particular, $N_{ \mathscr{ F}}(G)=N_{ \mathscr{ F}}(G,G)$. Let $N^i_{ \mathscr{ F}}(G)$,$i\geq 1$, be a upper series of $G$ by setting $N^0_{ \mathscr{ F}}(G)=1$, $N^{ i+1}_{ \mathscr{ F}}(G)/N^i_{ \mathscr{ F}}(G)=N_{ \mathscr{ F}}(G/N^i_{ \mathscr{ F}}(G))$ and denoted by $N^{ \infty}_{ \mathscr{ F}}(G)$ the terminal term of the series. In this paper, for the case $\mathscr{ F}\in\{ \mathscr{ U}_p,\mathscr{ U}\}$, where $\mathscr{ U}_p$($\mathscr{ U}$,respectively) is the class of all finite $p$-supersolvable groups(supersolvable groups,respectively), we characterize the structure of some given finite groups by the properties of weak norm of some subgroups in $G$ with respect to $\mathscr{ F}$. Some of our main results may regard as a continuation of many nice previous work.
Categories: math.GR